Second-order bulk-boundary correspondence in rotationally symmetric topological superconductors from stacked Dirac Hamiltonians

Two-dimensional second-order topological superconductors host zero-dimensional Majorana bound states at their boundaries. In this work, focusing on rotation-invariant crystalline topological superconductors, we establish a bulk-boundary correspondence linking the presence of such Majorana bound states to bulk topological invariants introduced by Benalcazar et al. [Phys. Rev. B 89, 224503 (2014)]. We thus establish when a topological crystalline superconductor protected by rotational symmetry displays second-order topological superconductivity. Our approach is based on stacked Dirac Hamiltonians, using which we relate transitions between topological phases to the transformation properties between adjacent gapped boundaries. We find that, in addition to the bulk rotational invariants, the presence of Majorana boundary bound states in a given geometry depends on the interplay between weak topological invariants and the location of the rotation center relative to the lattice. We provide numerical examples for our predictions and discuss possible extensions of our approach.

Figure 1

The Brillouin zone for $C_4$-symmetric models. There are two fourfold fixed points labeled $\mathit{\Gamma }$ and $\mathbf{M}$, and two twofold fixed points $\mathbf{X}$ and ${\mathbf{X}}^{\prime }$ that transform into each other upon a fourfold rotation. The shaded region indicates the fundamental domain that generates the entire BZ.

Figure 2

The rotation eigenvalues for $C_2$ and $C_4$ symmetry, respectively. The PH operator relates complex-conjugate pairs of rotation eigenvalues (while also switching between positive- and negative-energy bands).

Figure 3

Two options ($A$ and $B$) for the rotation center in an infinite $C_4$-symmetric lattice. The dotted box shows a primitive unit cell with its associated lattice site in the middle. Case $B$ has its rotation operator shifted by $\mathbf{c}=\frac{1}{2}({\mathbf{a}}_1+{\mathbf{a}}_2)$.

Figure 4

Example spectrum along the edge of a 2D superconductor. Each sign change of a bulk mass $m_{\alpha }$ manifests itself as a chiral mode on the edge. (a) The folding process which takes all HSPs to $\mathit{\Gamma }$, meaning that all edge modes are centered on $k_{\parallel }=0$ in the continuum description. (b) An edge mass term that couples a left mover to a right mover, opening up a gap on the edge.

Figure 5

Phase diagram for the lattice Hamiltonian (54). The black lines denote gap closings at $\mathit{\Gamma }$, the black dashed lines at $\mathbf{M}$, and the gray lines at $\mathbf{X},{\mathbf{X}}^{\prime }$. In (d), the black stripes in phases with $\mathrm{Ch}=0$ denote values of the topological index predicting corner modes. Diagonal stripes indicate ${\mathrm{{\rm Y}}}_{\mathbf{0}}^{\left(4\right)}=1$ and ${\mathrm{{\rm Y}}}_{({\mathbf{a}}_1+{\mathbf{a}}_2)/2}^{\left(4\right)}=0$, and a crossed pattern indicates ${\mathrm{{\rm Y}}}_{\mathbf{0}}^{\left(4\right)}={\mathrm{{\rm Y}}}_{({\mathbf{a}}_1+{\mathbf{a}}_2)/2}^{\left(4\right)}=1$. The gray square and black triangles mark the parameters used in Figs. 7 and 7, respectively.

Figure 6

Lattice model that demonstrates the importance of the rotation center. The Hamiltonian (54) at $\varphi =m=0$ only contains terms, illustrated by the solid blue lines, that couple neighboring Majorana modes, illustrated by the black circles; cf. Ref. [14]. In this case, Majorana modes at the edge are completely decoupled from the bulk and do not contribute to the Hamiltonian, i.e., all edge modes have zero energy. This degeneracy can be lifted by a density wave [14] modeled by coupling every second-nearest-neighbor site on the edge (dashed blue lines). (a) When the rotation center is in the center of a unit cell, any coupling that respects rotational invariance is bound to leave Majoranas at the corners uncoupled (red circles). This completely decoupled case is topologically equivalent to any case with a finite localization length of the corner modes, as for example considered in Fig. 7. (b) However, when the rotation center is at the corner of a unit cell, it is possible to open a surface gap without Majorana bound states.

Figure 7

Energy eigenvalues of the Hamiltonian defined in Eq. (54) with a boundary perturbation (with $t=0.025$) for a finite square lattice with $L\times L$ sites. The different colors denote the rotation eigenvalue $e^{i\pi /4(2p-1)}$ and the different symbols distinguish different rotation centers (crosses and circles corresponding to even and odd $L$, respectively). In (a), we show an example of phase I with $\varphi =\pi /32$ and $m=0.4$. The system only supports gapless corner modes for odd $L$, which corresponds to a physical rotation center in the center of a unit cell. In (b), we show an example of phase II with $\varphi =17\pi /32$ and $m=0.4$. The corner modes remain for both rotation centers, i.e., both even and odd $L$. We choose the logarithmic scale of the $y$ axis to visualize the exponential decrease in energy.

Figure 8

Different $C_4$-symmetric lattices. (a) Each unit cell respects $C_4$ symmetry individually and is compatible with a finite lattice. (b) Each unit cell respects $C_4$ symmetry, but contains fractional atomic sites and is therefore incompatible with a finite lattice. (c) For the same lattice as in (b), we can define a different unit cell that is compatible with a finite system, but does not respect $C_4$ symmetry.

Figure 9

(a) Brillouin zone for $C_2$-symmetric models. All HSPs are twofold fixed points, i.e., they map to themselves under a $C_2$ rotation. (b) Brillouin zone for $C_6$-symmetric models. Only $\mathit{\Gamma }$ is a sixfold fixed point, whereas the threefold fixed points $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ map to each other under a sixfold rotation. The twofold fixed points $\mathbf{M}$, ${\mathbf{M}}^{\prime }$, and ${\mathbf{M}}^{″}$ form an orbit $\mathbf{M}\to {\mathbf{M}}^{\prime }\to {\mathbf{M}}^{″}$ under sixfold rotation.